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A '''basis''' ''B'' for a topology ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that
''Basis is a [[mathematics]] term.''
== Linear algebra ==
In [[linear algebra]], a basis is a set of [[linear independence|linearly independent]] vectors that [[spans]] a vector space ''V''. Any vector in the vector space can then be written as a linear combination of the basis.
This usage of the term is similar to its common usage: a basis is the foundation for what is needed.
== Topology ==
A '''basis''' ''B'' for a [[topology]] ''T'' on a set ''X'' is a collection of subsets of ''X'' (called '''basis elements''') such that
#each ''x'' in ''X'', is in at least one basis element.
#each ''x'' in ''X'', is in at least one basis element.
#if x is in the intersection of 2 basis elements ''B<sub>1</sub>'' and ''B<sub>2</sub>'', then it is in some basis element ''B<sub>3</sub>'', where ''B<sub>3</sub>'' is a subset of ''B<sub>1</sub> ∩ B<sub>2</sub>''.
#if x is in the intersection of 2 basis elements ''B<sub>1</sub>'' and ''B<sub>2</sub>'', then it is in some basis element ''B<sub>3</sub>'', where ''B<sub>3</sub>'' is a subset of ''B<sub>1</sub> ∩ B<sub>2</sub>''.
If ''B'' satisfy the above 2 conditions, then the '''topology ''T'' generated by ''B''''' is the collection of subsets ''U'' of ''X'' such that for each ''x'' in ''U'', there is a basis element ''V'' in ''B'' such that ''x'' is in ''V'' and ''V'' is a subset of ''U''.
If ''B'' satisfy the above 2 conditions, then the '''topology ''T'' generated by ''B''''' is the collection of subsets ''U'' of ''X'' such that for each ''x'' in ''U'', there is a basis element ''V'' in ''B'' such that ''x'' is in ''V'' and ''V'' is a subset of ''U''.
[[category:topology]]
[[Category:Topology]][[Category:Linear Algebra]]